Let $\{f_n\}_n$ be a sequence of functions $\mathbb R \rightarrow \mathbb R$ which converges pointwise to $f$, ie:
$$f_n(x) \rightarrow f(x) \hspace{10pt}\hbox{for all $x$}.$$
What additional conditions are needed so that the derivatives at $0$ of $\{f_n\}_n$ converge to the derivatives at $0$ of $f$ ?
One condition that would make sense is "uniform convergence on all compact sets" though I can't seem to find references
Unlike integration, differentiation is a very unstable operation. It is very hard to make assumptions on $\{f_n\}_n$ so that $\{f'_n\}_n$ converges. For instance, let $f_n(x)= \frac{\sin (nx)}{n}$: $\{f_n\}_n$ converges to zero uniformly, but the derivatives $f'_n$ are oscillating.
The only "elementary" theorem about differentiation of sequences of functions assumes that $\{f'_n\}_n$ converges uniformly; the conclusion is that pointwise convergence (even at a single point) of $\{f_n\}_n$ implies uniform convergence.
All this is strictly related to the fact that the differentiation operator is unbounded as a linear operator in essentially every useful function space.